NCERT Solutions for Class 11 Maths Chapter 13 Limits and Derivatives

NCERT Solutions for Class 11 Maths Chapter 13 Limits and Derivatives are part of NCERT Solutions for Class 11 Maths. Here we have given NCERT Solutions for Class 11 Maths Chapter 13 Limits and Derivatives.

Board	CBSE
Textbook	NCERT
Class	Class 11
Subject	Maths
Chapter	Chapter 13
Chapter Name	Limits and Derivatives
Exercise	Ex 13.1, Ex 13.2
Number of Questions Solved	43
Category	NCERT Solutions

NCERT Solutions for Class 11 Maths Chapter 13 Limits and Derivatives

Chapter 13 Limits and Derivatives Exercise 13.1

Evaluate the following limits in Exercises 1 to 22.

Question 1.

Solution:

Question 2.

Solution:

Question 3.

Solution:

Question 4.

Solution:

Question 5.

Solution:

Question 6.

Solution:

Question 7.

Solution:

Question 8.

Solution:

Question 9.

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Question 10.

Solution:

Question 11.

Solution:

Question 12.

Solution:

Question 13.

Solution:

Question 14.

Solution:

Question 15.

Solution:

Question 16.

Solution:

Question 17.

Solution:

Question 18.

Solution:

Question 19.

Solution:

Question 20.

Solution:

Question 21.

Solution:

Question 22.

Solution:

Question 23.

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Question 24.

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Question 25.

Solution:

Question 26.

Solution:

Question 27.

Solution:

Question 28.

Solution:

Adding (ii) and (iii), we get 2b = 8 ⇒ b = 4
Subtituting the value of b in (iii), we get
4 – a = 4 ⇒ a = 0
Thus a = 0 and b = 4.

Question 29.

Solution:

Question 30.

Solution:

Question 31.

Solution:

Question 32.

Solution:

Chapter 13 Limits and Derivatives Exercise 13.2

Question 1.
Find the derivative of x² – ² at x = 10.
Solution:
let f(x) = x² – 2
Differentiating (i) with respect to x, we get
f'(x) = 2x
At x = 10, f'(10) = 2(10) = 20.

Question 2.
Find the derivative of 99x at x = 10.
Solution:
let f(x) = 99x
Differentiating (i) with respect to x, we get
f'(x) = 90
At x = 100, f'(100) = 99.

Question 3.
Find the derivative of x at x = 10.
Solution:
let f(x) = x
Differentiating (i) with respect to x, we get
f'(x) = 1
At x = 1, f'(1) = 1.

Question 4.
Find the derivative of the following functions from first principle.
(i) x³ – 27
(ii) (x – 1)(x – 2)
(iii)
(iv)
Solution:

Question 5.
For the function

Prove that f'(1) = 100f'(0)
Solution:
We have

Question 6.
Find the derivative of xⁿ + ax^n-1 + a²x^n-2+
…. + a^n-1x + aⁿ for some fixed real number a.
Solution:
Let f(x) = xⁿ + ax^n-1 + a²x^n-2+
…. + a^n-1x + aⁿ
Differentiating (i) with respect to x, we get
f'(x) = nx^n-1 + (n – 1)ax^n-2 + …… + a^n-1

Question 7.
For some constants a and b, find the derivative of
(i) (x – a)(x – b)
(ii) (ax² + b)²
(iii)
Solution:
(i) Let f(x) = (x – a)(x – b) ….(1)
Differentiating (1) with respect to x, we get
f'(x) = (x – a)(x – b)’ + (x – a)’ (x – b)
⇒ f'(x) = (x – a) + (x – b) = 2x – a – b

Question 8.
Find the derivative for some constant a.
Solution:
Let f(x) = ….(i), where a is a constant.
Differentiating (i) with respect to x, we get

Question 9.
Find the derivative of

Solution:
(i) Let f(x) = …(1)
Differentiating (i) with respect to x, we get
f'(x) = 2·1 – 0 ⇒ f'(x) = 2.
(ii) Let f(x) = 5x³ + 3x – 1)(x – 1)

Question 10.
Find the derivative of cos x from first principle.
Solution:
Let f(x) = cos x

Question 11.
Find the derivative of the following functions:
(i) sin x cos x
(ii) secx
(iii) 5 secx + 4 cosx
(iv) cosecx
(v) 3 cotx + 5 cosecx
(vi) 5sinx – 6 cosx + 7
(vii) 2 tanx – 7 secx.
Solution:
(i) Let f(x) = sin x cos x … (1)
Differentiating (1) with respect to x, we get

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